![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rspsn | Structured version Visualization version GIF version |
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rspsn.b | ⊢ 𝐵 = (Base‘𝑅) |
rspsn.k | ⊢ 𝐾 = (RSpan‘𝑅) |
rspsn.d | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
rspsn | ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2832 | . . . . 5 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
3 | 2 | rexbidv 3262 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
4 | rlmlmod 19566 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
5 | rlmsca2 19562 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
6 | baseid 16282 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
7 | rspsn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | strfvi 16276 | . . . . 5 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
9 | rlmbas 19556 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
10 | 7, 9 | eqtri 2849 | . . . . 5 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
11 | rlmvsca 19563 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | rspsn.k | . . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅) | |
13 | rspval 19554 | . . . . . 6 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
14 | 12, 13 | eqtri 2849 | . . . . 5 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
15 | 5, 8, 10, 11, 14 | lspsnel 19362 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
16 | 4, 15 | sylan 577 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
17 | rspsn.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
18 | eqid 2825 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 7, 17, 18 | dvdsr2 19001 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
20 | 19 | adantl 475 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
21 | 3, 16, 20 | 3bitr4d 303 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝐺 ∥ 𝑥)) |
22 | 21 | abbi2dv 2947 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {cab 2811 ∃wrex 3118 {csn 4397 class class class wbr 4873 I cid 5249 ‘cfv 6123 (class class class)co 6905 ndxcnx 16219 Basecbs 16222 .rcmulr 16306 Ringcrg 18901 ∥rcdsr 18992 LModclmod 19219 LSpanclspn 19330 ringLModcrglmod 19530 RSpancrsp 19532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-mgp 18844 df-ur 18856 df-ring 18903 df-dvdsr 18995 df-subrg 19134 df-lmod 19221 df-lss 19289 df-lsp 19331 df-sra 19533 df-rgmod 19534 df-rsp 19536 |
This theorem is referenced by: lidldvgen 19616 zndvds 20257 |
Copyright terms: Public domain | W3C validator |